Optimization: Difference between revisions
Merge basic concepts links into text body, formatting, misc. edits |
Clarify. Hopefully this defends the terminology choice here |
||
| Line 4: | Line 4: | ||
== Taxonomy == | == Taxonomy == | ||
Roughly speaking, there are two types of tunings with diverging philosophies: prime-based tunings and target tunings. | Roughly speaking, there are two types of tunings with diverging philosophies: ''prime-based tunings'' and ''target tunings''. | ||
* A prime-based tuning is optimized for the [[formal prime]]s, but they are representative for the set of all intervals. There are two perspectives. First, in the [[Vals and tuning space|tuning space]], it minimizes the errors of formal primes. Second, in the [[Monzos and interval space|interval space]], it rates all intervals through a norm, which serves as a complexity measure, and it minimizes the maximum [[damage]] (i.e. error divided by complexity) for all intervals. | * A prime-based tuning is optimized for the [[formal prime]]s, but they are representative for the set of all intervals. There are two perspectives. First, in the [[Vals and tuning space|tuning space]], it minimizes the errors of formal primes. Second, in the [[Monzos and interval space|interval space]], it rates all intervals through a norm, which serves as a complexity measure, and it minimizes the maximum [[damage]] (i.e. error divided by complexity) for all intervals. | ||
* A target tuning is | * A target tuning is optimized for a particular set of intervals and considers the rest irrelevant. However, the interval does not get infinite complexity even if it is disregarded due to the normed nature of the interval space, so these tunings also correspond to all-interval damage minimizations of some sorts. | ||
This article focuses on prime-based tunings. See the dedicated page (→ [[Target tunings]]) for target tunings. | This article focuses on prime-based tunings. See the dedicated page (→ [[Target tunings]]) for target tunings. | ||
| Line 25: | Line 25: | ||
An orthogonal space treats divisive ratios as equally important as multiplicative ratios, yet divisive ratios are sometimes thought to be more important. For example, 5/3 is sometimes found to be more important than 15. The skew is introduced to address that. | An orthogonal space treats divisive ratios as equally important as multiplicative ratios, yet divisive ratios are sometimes thought to be more important. For example, 5/3 is sometimes found to be more important than 15. The skew is introduced to address that. | ||
Notably, adopting the [[Weil height]] will skew the space | Notably, adopting the standard [[Weil height]] will skew the space to 60 degrees. | ||
Both the weight and the skew are represented by matrices that can be applied to the mapping. In a more general sense, the distinction may not matter, and they may be collectively called by either part. | Both the weight and the skew are represented by matrices that can be applied to the mapping. In a more general sense, the distinction may not matter, and they may be collectively called by either part. | ||
| Line 36: | Line 36: | ||
The Minkowskian norm, Manhattan norm or taxicab norm aka ''L''<sup>1</sup> norm resembles movement of taxicabs in Manhattan – it can only traverse horizontally or vertically. A diagonal movement mounts to two steps. | The Minkowskian norm, Manhattan norm or taxicab norm aka ''L''<sup>1</sup> norm resembles movement of taxicabs in Manhattan – it can only traverse horizontally or vertically. A diagonal movement mounts to two steps. | ||
The Chebyshevian norm aka ''L''<sup> | The Chebyshevian norm aka ''L''<sup>infinity</sup> norm is the opposite of the Minkowskian norm – it is the maximum number of steps in any direction, so a diagonal movement is the same as a horizontal or vertical one. | ||
It should be noted that the dual norm of ''L''<sup>1</sup> is ''L''<sup>infinity</sup>, and vice versa. Thus, the Minkowskian norm corresponds to the ''L''<sup>infinity</sup> tuning space, and the Chebyshevian norm corresponds to the ''L''<sup>1</sup> tuning space. The dual of ''L''<sup>2</sup> norm is itself, so the Euclidean norm corresponds to Euclidean tuning as one may expect. | It should be noted that the dual norm of ''L''<sup>1</sup> is ''L''<sup>infinity</sup>, and vice versa. Thus, the Minkowskian norm corresponds to the ''L''<sup>infinity</sup> tuning space, and the Chebyshevian norm corresponds to the ''L''<sup>1</sup> tuning space. The dual of ''L''<sup>2</sup> norm is itself, so the Euclidean norm corresponds to Euclidean tuning as one may expect. | ||